Stiffness of a tube is determined by two things: the shape factor called "area moment of inertia" and the metals stiffness as denoted by the "modulus of elasticity". You didn't provide enough information to answer the question since these two factors can be juggled between them to build a frame of various characteristics.

Yes, assuming the same wall thickness Aero tube have two advantages, the narrow axis is flatter surfaced, which can mean a lot more surface area to carry the load, like an Ibeam with a wide flange, so it will be stiffer and stronger across the minor axis. And the deeper axis is also going to be stiffer since the stiffness varies to the cube of the diameter, stronger to the square, though the stress may be higher given the pointier shape. Given these benefits, the same diameter or wall thickness tube is not likely used and then you are back to N's answer.

At high speed the streamlined tube is far more aero efficient, so it's size is mostly a weight handicap. This is why you can gain a lot by enclosing a human in an aero fairing even if it has greater frontal area and surface area.

Those aero chromo tubes are designed for wing struts and are heavy walled, at 200 mph the whole wing on a cessna has the same drag as a 3/8" rod of the same length, according to Hoyt.

Stiffness is an easy thing to measure you can just try bending/flexing both tubes and see where it gets you. Try sitting on a 2x4 flat across two posts, on edge vs on flat. Your question is like comparing a 2x4 to a 2x2. A pure oval which is just a stretched circle, could be expected (I am guessing) to be stiffer across the narrow axis to the extent the longer axis is stretched. So the 2x1 oval would be twice as stiff on the narrow axis vs 1x1 circle, and 8 times as stiff on the deep axis (2 cubed, vs 1 cubed). This stuff rapidly moves to higher pay grades than mine, but I don't think I would be far off on those models. Amazing what you can find out with a few sticks and some weights or a spring fishing scale.

Important, to compare the deflection rates also. Just keep in mind that what you discover over a 1% bend may not help you if your application is a 10% bend. In other words, what you want in a structure for a bow is not the same thing as a joist.

Oval tubes are stiffer against flex in the direction of the major axis, and more flexible in the direction of the minor axis compared to a round tube.

In a bicycle frame for example, an aero down tube will make the frame stiff in the vertical direction (frame will ride very stiff) yet the bottom bracket area of the frame will be flexi side to side. Not the best combination in my opinion.

Yes, the frame will be more aerodynamic, but aero when it comes to the frame is a very minor consideration for all but elite cyclists running time trials. The riders position on the bike is most important followed by wheel drag. Drag from a frame is almost nothing compared to these two factors so I wouldn't worry about it.

"Oval tubes are stiffer against flex in the direction of the major axis, and more flexible in the direction of the minor axis compared to a round tube."

Yeah I guess even this didn't have adequate info. I assume he means comparison to tube with same size minor axis, like a 1 x 1.7 aero tube, vs. a 1 in round tube. Same wall thickness the aero tube is stiffer and strong in all axis. However if he is thinking 1 inch round compared to 5/8 x1" aero, that would be another mater. If we are talkng aero cromo like spruce sells it's heavy stiff and might still be in the running.

The aero is pretty significant, per tube, but how significant any one tube is, is another question.

That's why at some level it is just easier to do the bend testing. In archery we spine every arrow. In aircraft we rate wood in flex jigs.

Bending stiffness : depends on EI (tensile modulus and moment of inertia)
Torsional stiffness : depends on GJ (polar moment of inertia and shear modulus, which can also be written in terms of Poisson's ratio with tensile modulus)
Axial stiffness : depends on EA or KA (tensile/compressive modulus and area of x-section). Usually this is the least important one.

Steel is the stiffest frame material (besides carbon), and aluminum alloy the highest specific stiffness (10% greater than steel or titanium). Titanium offers a combination of light weight and stiffness, but you can say that it's good at both, sucks at both, etc.

The tubes in a "typical" bicycle frame don't flex against an axis, per se. Most stress on a bike is torsional, and for that a round tube is better than a shaped tube. Don't buy into the marketing hype. Most "aero" tubes aren't actually aero and ovalized tubes are ovalized to make the construction of the joints easier, not to make the bike stiffer.

The tubes in a "typical" bicycle frame don't flex against an axis, per se. Most stress on a bike is torsional, and for that a round tube is better than a shaped tube. Don't buy into the marketing hype. Most "aero" tubes aren't actually aero and ovalized tubes are ovalized to make the construction of the joints easier, not to make the bike stiffer.

Torsion accounts for roughly 65% of the stress in the down tube and the rest is bending. An oval tube with the major axis sideways at the bottom bracket is the ideal shape to resist deflection since the oval shape handles torsion well and with the major axis of the oval resisting bending.

The tubes in a "typical" bicycle frame don't flex against an axis, per se. Most stress on a bike is torsional, and for that a round tube is better than a shaped tube. Don't buy into the marketing hype. Most "aero" tubes aren't actually aero and ovalized tubes are ovalized to make the construction of the joints easier, not to make the bike stiffer.

As Peter mentioned, the loads on each tube is different. Columbus had a tubeset with each one shaped differently to deal with its unique loads. Forgot the name.. was it OX?

Another thing that wasn't mentioned was the exact wall-thickness of the tubing in question. What if you were to compare round versus ovalized vs. square tubing of the same weight, how would their stiffness in different directions compare?

If by that you mean that oval isn't the perfect aero shape, it is a whole lot better than round. Ovals actualy get pretty close to wing shapes. I know one boat that after years replaced a really aero section with a carbon spar that was merely oval, but due to carbon it was a little thinner. That was more decisive in performance than the section. Not a lot of specifics, but it does show how easy it is for different factors to gain traction versus idealized concepts.

As Peter mentioned, the loads on each tube is different. Columbus had a tubeset with each one shaped differently to deal with its unique loads. Forgot the name.. was it OX?

Another thing that wasn't mentioned was the exact wall-thickness of the tubing in question. What if you were to compare round versus ovalized vs. square tubing of the same weight, how would their stiffness in different directions compare?